Univariate Frequency Distributions , Measures of Central Tendency : Mean , Median and Mode , Arithmetic , Geometric and Harmonic Mean . Measures of dispersion , Skewness and Kurtosis

 

Forming a Frequency Distribution : A frequency distribution is a list , table or graph that displays the frequency of various outcomes in a sample. Each entry in the table contains the frequency or count of the occurrence of values within a particular group or Interval. 

Considerations while Forming a Frequency Distribution:- 

1. Appropriate number of classes 
2. Choice of class mid points 
3. Continuous(this amount and under that amount) or Discrete variable (this amount to that amount) 
4. Width of the class be measured by the difference between lower (or upper) limits of two adjacent classes and not by the difference between the lower and upper limit of a given class. 

Relative Frequency Distribution is used to show relative frequencies rather than absolute frequencies. The relative frequencies must add up to 100%.

Types of Frequency Distribution 

1. Symmetrical 
2. Positively Skewed 
3. Negatively Skewed 

Measure of Central Value - Summary measures to describe frequency distribution or representative values at which the distribution is centred. The symbol Σ means the 'sum of'. Thus if X consists of three values , eg 4 , 7 and 9 , ΣX instructs us to sum all the values of X. 

       ΣX= 4+7+9= 20 

• For clarity  , we sometimes attach subscripts to the individuals values of the variable X , i.e., X1= 4 , X2= 7 ,X3= 9. Then we write   Σ^3{i=1} = X1+X2+X3= 20 .

Computing  Mean From Grouped Data 

 Method 1 : Writing X for class mid points and f for the frequencies in the classes , the mean for grouped data is defined as 

                           Mean= ΣfX/Σf= (ΣfX)/N , Where N=Σf. 

Class mid point = Lower limit + Upper Limit / 2

Method 2 : X' = (A + Σf)X/N where X' is the deviation of a class mid- point from an arbitrary origin .

Method 3 : X' = A +( ΣfX'/N*h) , where X' is in class interval units and h is the width of the class interval. 

 Class Width, h = Upper Limit - Lower Limit 

 Weighted Arithmetic Mean 

The mean for a frequency distribution , X'=ΣfX/Σf , is in fact a mean of the X's(class mid-points) where each X is weighted by its importance. This is only a special case of the more general notion of a weighted mean , X' = ΣWX/ΣW , where the Ws are weights. 

The concept of weighted mean can be simplified by expressing the given weights as relative weights V = W/ΣW so that ΣW=1. Then X'=ΣVX. 

Example- Suppose that bread is sold at three prices: 17 cents , 19 cents and 22 cents a loaf. The simple mean price is 19.3 cents but a more useful measure is the mean which results from attaching  to each price the quantity of bread sold at that price ,i.e.,  a weighted mean. If the sales per week the three type of bread are 50k , 30k and 20k loaves respectively , we should have 

               X' = (17 x 50,000 + 19 x 30,000 + 22 x 20,000)/100,000 = 18.6 cents 

Using relatives weighted this reduces to  

            X' = (17 x 0.5 ) + (19 x 0.3 ) + (22 x 0.2 ) = 18.6 cents .